‘This is a false statement’: is something beyond our logical reasoning? ― A review of Douglas R. Holfstadter’s Gödel, Escher, BachRead Now
‘A good poet will usually borrow from authors remote in time, or alien in language, or diverse in interest.’ ― T.S. Eliot, The Sacred Wood
Douglas R. Holfstadter has made an unprecedented move to bond the mathematician Kurt Gödel, the artist Maurits Corpelis Escher, and the composer Johann Sebastian Bach in the common centre of Gödel’s proof on his incompleteness theorem. This proof is targeting the intrinsic logical looseness and limitations in a formal axiomatic system in mathematics.
In a nutshell, mathematicians have been fancying exploiting a list of axioms which could give us “all of the mathematics”: it has to be a complete system – that any given statement is both provable and disprovable; it also has to be a consistent system – that a statement cannot be both proved true and false at the same time. However, such completeness and consistency are inherently contradictory – for example, ‘This statement does not have any proof in the system of Principia Mathematica (PM)’: if this statement is provable, then PM would be inconsistent (PM is self-referentially contradicting); if this statement is unprovable, then PM would be incomplete (the PM lacks the internal proof for it).
The mathematical ground of Gödel’s proof itself is a stand-alone masterpiece, but it also steps further to reveal the epistemology of any formal system science – theorems are the branch-outs developed from an axiomatic trunk, extending towards the vast space of truth while some being unreachable, at its counterpart, negative axioms provide the basis of all negations of theorems, also leaving some falsehoods unreachable. Coincidently, this reminds me a Chinese counterpart: Yin Yang Theory, and the symbol itself would give you the intuitive thought of its similarity with Gödel’s proof (Figure 1).
The book is precious in the presentation of such rigorous and complex ideas. First, the dialogue at the beginning of each chapter unveils the limitations that we may come across in our daily lives – such opening encores with the dialogues in Plato’s Symposium. It successfully rings a bell among the readers, preparing the laymen to digest the mathematical and logical paradoxes. Second, the author’s imagination is far beyond mathematics alone: he traces back the connections from Escher and Bach with the Gödel’s theorem, offering a highly vibrant repertoire of aesthetics that is engraved in any field of knowledge. For example, Bach’s canons and fugues are often self-referential to deliver ambiguous perceptions to the listeners. In the meanwhile, many of the Escher’s artwork seem unreal and challenge our intuitive perception of space. Third, at the final portion of the book, it brings out a twenty-first century grand challenge of artificial intelligence, and how it may shed light on resolving the complex systems of self-reference, offering valuable insight to the futurism.
This book is definitely a classic that any learners at all stages of knowledge and truth inquiry should read, for the most underlying structure of how we learn and perceive.
T. H. (Henry) Hung